A certain brewery uses an automated process to fill their bottles with 12 ounces of beer. They want to test the machines to see if they are indeed filling to 12 oz on average. They plan to sample 12 bottles off the line and measure their actual fill level. They decide that if the average fill level from the sample is more than 0.05 ounces above or below the desired 12 oz, they will have the machines recalibrated. Use s= 0.2, and assume that n= 12 is sufficiently "large" enough to use the CLT.
(a) What's the probability that they will decide to have the machines recalibrated if in fact the machines are working properly? (i.e. what is the probability that the sample mean is more than 0.05 from 12, if µ= 12?)
(b) If the truth is that µ= 11.9 (the machines are under filling), what's the probability that they (correctly) have the machines recalibrated?
(c) What would the cutoff need to be (how far above or below 12 would the sample mean have to be) for the error probability in part (a) to be 0.01?
(d) Using the new cutoff from the previous part, what is the probability that they will decide to have the machines recalibrated if the truth is that µ= 11.9?
(e) How do the probabilities in parts (b) and (d) compare? Why is this?